
Bilinear form is defined as:
\(\min _{u\in \mathbb {R}^p}\max _{v\in \mathbb {R}^q} ~u^{\top } Av+u^{\top }a+v^{\top } b,\) 
where \(a\in \mathbb {R}^p, b\in \mathbb {R}^q\)  and \(A\in \mathbb {R}^{p\times q}\) . Bilinear setting is often used as toy examples in min-max optimization.

Let \(K:\mathbb {R}^d\rightarrow \mathbb {R}^d\)  be an operator and \(\mathcal {X}\subseteq \mathbb {R}^d\)  is a closed convex domain. [1]} proposed the Stampacchia Variational Inequality (SVI), which aims to find \(z^*\in \mathcal {X}\) , such that \(\langle K(z^*), z-z^*\rangle \geqslant 0\)  holds for all \(z\in \mathcal {X}\) . Similarly, [2]} proposed the Minty Variational Inequality (MVI) problem, which aims to find \(z^*\in \mathcal {X}\) , such that \(\langle K(z), z-z^*\rangle \geqslant 0\) . In Table REF , the operator \(K(z)=(\nabla _x \phi (x,y), -\nabla _y \phi (x,y))^{\top }=-V(z)\)  with \(z=(x,y)\) .

